> < ^ Date: Mon, 04 Mar 2002 10:18:44 +0000
> < ^ From: John J McDermott <jjm@dcs.st-and.ac.uk >
< ^ Subject: Re: Metaplektische Gruppe

Dear Forum,

The following is forwarded from Rudy Beyl (rudy@mth.pdx.edu).

Lieber Herr Freitag, dear GAP-Forum:

<freitag@mathi.uni-heidelberg.de> wrote in GAP Forum article 3045:

>the group SL(2) over the ring Z/qZ (for example q=12 is interesting)
>admits a twofold covering, the mataplectic group.

<wdj@usna.edu> wrote in GAP Forum article 3047:

>If memory serves, when q is prime a thrm of Steinberg implies that
>such a covering splits. GAP's DirectProduct command ...
>Do you know how to define the cocycle on SL(2,Z/qZ) when q is not
>a prime? - David Joyner

Assuming that "covering" means "central stem extension" or
"representation group", a.k.a. "stem cover", I concur:
SL(2,Z/qZ) has no proper central stem extension and every complex
projective representation can be linearized, when q is not divisible by 4,
since in all these cases the Schur multiplicator of SL(2,Z/qZ) is trivial.
However, as SL(2,Z/qZ) is not perfect when q is divisible by 2 or 3, there
exist non-split central extensions by SL(2,Z/qZ) .

<freitag@mathi.uni-heidelberg.de> wrote in GAP Forum article 3048:

>The interesting case is that 4 divides q. Then there is a covering
>which does not split. The cocycle can be computed.

(Joachim Neubueser and) <max.neunhoeffer@math.rwth-aachen.de> wrote
in GAP Forum article 3050:

>There are in fact two groups that qualify as
>twofold covers of SL(2,Z/12Z): ...

Indeed, whenever q is divisible by 4, SL(2,Z/qZ) has two non-isomorphic
stem covers CC_q and CK_q , say, both are non-split central stem
extensions. The first has central elements of order 4 , the center of the
other is elementary 2-abelian. I have no idea which of these would
qualify as "the metaplectic group".

If q = k*m , where k is a power of 2 and m is odd, then SL(2,Z/qZ) , CC_q ,
and CK_q are the direct product of SL(2,Z/mZ) and SL(2,Z/kZ) , CC_k ,
and CK_k , respectively. The center of CC_k is Z/4Z, of CK_k the Klein
four group. For large q, these facts will reduce workspace problems in
any computation.

The above is extracted from a paper of mine, [Math. Z. 191 (1986), 23-42;
MR 87b:20071, Zbl 581.20050].  Section 4 of that paper has free
presentations for these groups.  In particular, CC_4 of order 96 has the
beautiful presentation < a, b : a^4 , (a*b)^3*b^-2 > and CC_12 is the
direct product of CC_4 and the binary tetrahedral group SL(2,Z/3Z) .

F. Rudolf Beyl
Department of Mathem. Sciences       Tel. 503-725-3646 (answered by me
Portland State University              or voice-mail recording device)
P.O. Box 751                         FAX: 503-725-3661
Portland, OR 97207-0751              Internet: beylf@pdx.edu

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